document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=624262>Question 624262</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a^2-7a-18/4a^3 divided by a^2-4a-45/2a^3-4a^2 times a^2+3a-10/a^2-4a+4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #392646 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=392646\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28a%5E2-7a-18%29%2F%284a%5E3%29%29%2F%28%28a%5E2-4a-45%29%2F%282a%5E3-4a%5E2%29%29%2A%28%28a%5E2%2B3a-10%29%2F%28a%5E2-4a%2B4%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "To divide fractions we learn to change it to multiplication of the reciprocal. So this becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28a%5E2-7a-18%29%2F%284a%5E3%29%29%2A%28%282a%5E3-4a%5E2%29%2F%28a%5E2-4a-45%29%29%2A%28%28a%5E2%2B3a-10%29%2F%28a%5E2-4a%2B4%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Back in the \"good old days\", before we had variables in the fractions, we learned that, when you are multiplying fractions,<ul><li>it was OK to cancel <i>before</i> multiplying the fractions.</li><li>it was OK to \"cross-cancel\" (i.e. cancel a factor of one fraction's numerator with a factor of another fraction's denominator. For example:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2A%283%2F5%29%2A%282%2F7%29%2A%285%2F11%29 ALIGN=MIDDLE  ALT=\"%281%2F2%29%2A%283%2F5%29%2A%282%2F7%29%2A%285%2F11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Fcross%282%29%29%2A%283%2Fcross%285%29%29%2A%28cross%282%29%2F7%29%2A%28cross%285%29%2F11%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F1%29%2A%283%2F1%29%2A%281%2F7%29%2A%281%2F11%29 ALIGN=MIDDLE  ALT=\"%281%2F1%29%2A%283%2F1%29%2A%281%2F7%29%2A%281%2F11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F77 ALIGN=MIDDLE  ALT=\"3%2F77\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li><li>that this \"pre-multiplication\" canceling made the multiplication easier and the post-multiplication fraction easier to reduce.</li></ul>Well not only is this still true with more complicated fractions, the advantages of \"pre-multiplication\" canceling are more important than ever!! So before we even think about multiplying these fractions together we want to cancel as many factors as we can. And to cancel factors we have to know what they are. So the next step is to factor each numerator and denominator as much as possible:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-7a-18 ALIGN=MIDDLE  ALT=\"a%5E2-7a-18\"   DISABLED_style= \"max-width:90%; height:auto;\" > factored as a trinomial: (a-9)(a+2)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4a%5E3 ALIGN=MIDDLE  ALT=\"4a%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is already factored although we could factor the 4 and \"factor out the exponent\": <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A2%2Aa%2Aa%2Aa ALIGN=MIDDLE  ALT=\"2%2A2%2Aa%2Aa%2Aa\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E3-4a%5E2 ALIGN=MIDDLE  ALT=\"2a%5E3-4a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > has a common factor of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2 ALIGN=MIDDLE  ALT=\"2a%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%5E2%28a-2%29 ALIGN=MIDDLE  ALT=\"2a%5E2%28a-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or 2*a*a(a-2)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-4a-45 ALIGN=MIDDLE  ALT=\"a%5E2-4a-45\"   DISABLED_style= \"max-width:90%; height:auto;\" > factored as a trinomial: (a-9)(a+5)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2%2B3a-10 ALIGN=MIDDLE  ALT=\"a%5E2%2B3a-10\"   DISABLED_style= \"max-width:90%; height:auto;\" > factored as a trinomial: (a+5)(a-2)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-4a%2B4 ALIGN=MIDDLE  ALT=\"a%5E2-4a%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > factored as a trinomial (or with the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2-1ab_b%5E2+=+%28a-b%29%28a-b%29 ALIGN=MIDDLE  ALT=\"a%5E2-1ab_b%5E2+=+%28a-b%29%28a-b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > pattern): (a-2)(a-2)<BR>\n" );
document.write( "Rewriting<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28a%5E2-7a-18%29%2F%284a%5E3%29%29%2A%28%282a%5E3-4a%5E2%29%2F%28a%5E2-4a-45%29%29%2A%28%28a%5E2%2B3a-10%29%2F%28a%5E2-4a%2B4%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "with factored numerators and denominators:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28%28a-9%29%28a%2B2%29%29%2F%282%2A2%2Aa%2Aa%2Aa%29%29%2A%28%282%2Aa%2Aa%28a-2%29%29%2F%28%28a-9%29%28a%2B5%29%29%29%2A%28%28%28a%2B5%29%28a-2%29%29%2F%28%28a-2%29%28a-2%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "We should see that we have quite a few factors to cancel out!<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28cross%28%28a-9%29%29%28a%2B2%29%29%2F%28cross%282%29%2A2%2Across%28a%29%2Across%28a%29%2Aa%29%29%2A%28%28cross%282%29%2Across%28a%29%2Across%28a%29cross%28%28a-2%29%29%29%2F%28cross%28%28a-9%29%29cross%28%28a%2B5%29%29%29%29%2A%28%28cross%28%28a%2B5%29%29cross%28%28a-2%29%29%29%2F%28cross%28%28a-2%29%29cross%28%28a-2%29%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "leaving:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28a-9%29%2F%282a%29%29%2A%281%2F1%29%2A%281%2F1%29 ALIGN=MIDDLE  ALT=\"%28%28a-9%29%2F%282a%29%29%2A%281%2F1%29%2A%281%2F1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which, you must agree, is much easier to multiply than<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28a%5E2-7a-18%29%2F%284a%5E3%29%29%2A%28%282a%5E3-4a%5E2%29%2F%28a%5E2-4a-45%29%29%2A%28%28a%5E2%2B3a-10%29%2F%28a%5E2-4a%2B4%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Multiplying out reduced fractions we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a-9%29%2F%282a%29 ALIGN=MIDDLE  ALT=\"%28a-9%29%2F%282a%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If we fully factored and canceled, then there should be no need to reduce after multiplying. And as you can see, there are no factors that will cancel here. (The \"a\" in the numerator is not a factor so you cannot cancel it with the \"a\" factor in the denominator.) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );